Electromagnetic Radiation (Waves)

We can manipulate Maxwell's equations such that we arrive at two concise expressions that relate spatial and temporal derivatives of the electric and magnetic fields. These are known as the wave equation (either for the magnetic or electric field):

$LaTeX: \nabla ^2\overset{\rightharpoonup }{E}=\epsilon _0\mu _0\frac{\partial ^2\overset{\rightharpoonup }{E}}{\partial t^2}$ $\nabla ^2\overset{\rightharpoonup }{E}=\epsilon _0\mu _0\frac{\partial ^2\overset{\rightharpoonup }{E}}{\partial t^2}$

and

$LaTeX: \nabla ^2\overset{\rightharpoonup }{B}=\epsilon _0\mu _0\frac{\partial ^2\overset{\rightharpoonup }{B}}{\partial t^2}$ $\nabla ^2\overset{\rightharpoonup }{B}=\epsilon _0\mu _0\frac{\partial ^2\overset{\rightharpoonup }{B}}{\partial t^2}$

where for instance, in Cartesian coordinates

$LaTeX: \nabla ^2=\partial _x^2+\partial _y^2+\partial _z^2$ $\nabla ^2=\partial _x^2+\partial _y^2+\partial _z^2$

and the Laplacian operates on each component of the fields separately

$LaTeX: \nabla ^2\overset{\rightharpoonup }{E}=\nabla ^2E_x\hat{i}+\nabla ^2E_y\hat{j}+\nabla ^2E_z\hat{k}$ $\nabla ^2\overset{\rightharpoonup }{E}=\nabla ^2E_x\hat{i}+\nabla ^2E_y\hat{j}+\nabla ^2E_z\hat{k}$

this means that we actually have three equations for the electric and magnetic fields. Again, in Cartesian coordinates, we would have (for the electric field)

$LaTeX: \partial _x^2E_x+\partial _y^2E_x+\partial _z^2E_x=\epsilon _0\mu _0\partial _t^2E_x ,$ $\partial _x^2E_x+\partial _y^2E_x+\partial _z^2E_x=\epsilon _0\mu _0\partial _t^2E_x ,$

$LaTeX: \partial _x^2E_y+\partial _y^2E_y+\partial _z^2E_y=\epsilon _0\mu _0\partial _t^2E_y ,$ $\partial _x^2E_y+\partial _y^2E_y+\partial _z^2E_y=\epsilon _0\mu _0\partial _t^2E_y ,$

and

$LaTeX: \partial _x^2E_z+\partial _y^2E_z+\partial _z^2E_z=\epsilon _0\mu _0\partial _t^2E_z$ $\partial _x^2E_z+\partial _y^2E_z+\partial _z^2E_z=\epsilon _0\mu _0\partial _t^2E_z$

there are several things to note about these equations. First, all of these equations have the form:

$LaTeX: \partial _x^2\psi +\partial _y^2\psi +\partial _z^2\psi =\frac{1}{v^2}\partial _t^2\psi$ $\partial _x^2\psi +\partial _y^2\psi +\partial _z^2\psi =\frac{1}{v^2}\partial _t^2\psi$

provided that

$LaTeX: v=\frac{1}{\sqrt{\mu _0 \epsilon _0}}$ $v=\frac{1}{\sqrt{\mu _0 \epsilon _0}}$ .

It was of great historical importance that Maxwell calculated $LaTeX: v=\frac{1}{\sqrt{\mu _0 \epsilon _0}} = 3\times10^8 m/s$ $v=\frac{1}{\sqrt{\mu _0 \epsilon _0}} = 3\times10^8 m/s$ , and noted the number was in good agreement with measurements made by Fizeau for the speed of light. This indicates that the electric and magnetic fields we have been working with are in fact constituent components of light and other forms of radiation. We use the symbol 'c' for the speed of light, it comes from Latin word celer, which means fast.

Another point to make about all of these equations is that each of the components of the electric and magnetic fields (i.e., LaTeX: E_x, E_y, E_z $E_x, E_y, E_z$ ) evolve independently when ρ = 0 and $LaTeX: \overset{\rightharpoonup }{J}=0$ $\overset{\rightharpoonup }{J}=0$ . Finally, as discussed previously, light obeys the principal of superposition, which means that light adds (and subtracts) linearly with itself.